One of the most important tasks of neuroscience is modeling the human brain. This is a very difficult task, because the human brain is incredibly complex and consists of more than 80 billion neurons. Modeling it would make it possible to make a big step in studying the features of brain activity, in the treatment of certain diseases and in understanding many processes in the human body.However, this task has not been solved yet.
To study the dynamics of processes in the nervous system, a FitzHugh-Nagumo model has been developed. It is a system of second-order ordinary differential equations and is one of the most common mathematical models of a nerve cell (neuron). Scientists around the world use it to model and study not only the dynamics of a single neuron, but also neural populations.

There are works in which researchers are able to recreate the electrical activity of an entire human brain accurately, for example, during an epileptic seizure, using a network of only 90 such models.

“It seems to us that we can go even further if we use the tools of control theory. The fact is that it is not at all obvious which parameters of the FitzHugh-Nagumo model should be chosen so that it should reflect the dynamics of a nerve cell plausibly. We propose to use an algorithm based on the application of the speed gradient method and differentiator filters to solve this problem, known as the identification problem. All you need to use it is to measure the values of voltage, membrane potentials of neurons," said Alexandra Rybalko, an intern researcher at the IPMash RAS laboratory Digitalization, Analysis and Synthesis of Complex Mechanical Systems, Networks and Media, who is working under the supervision of A.L.Fradkov, Professor, C.S.O.

An important result of this approach is that its complexity does not increase when the number of simulated neurons changes. That is, the task of modeling a network with tens of billions of nodes, as in the case of the human brain, no longer seems so unattainable.

In addition, the scientists managed to prove mathematically that, depending on these data, the model parameters will be adjusted accurately, that is, the dynamics of the model with these parameters will repeat the dynamics of a real neuron or a neural population. It is equally important that the approach takes into account measurement errors which arise due to imperfections in the equipment and can have a significant impact on the performance of the model.

“In the future, it is planned to apply our result to solve the problems of classifying brain activity modes, that can be used both for the study of diseases such as epilepsy and attention deficit hyperactivity disorder (ADHD), and for the development of brain-computer interfaces which allow robots to be controlled by the «power of thought» and are used actively in the field of neuroprosthetics," said Alexandra Rybalko.

Alexandra Rybalko's report on the results of this study Identification of Two-Neuron FitzHugh-Nagumo Model Based on the Speed-Gradient and Filtering was recognized as the best in the section Information Technology and Mathematics of the youth scientific forum Science of the Future — Science of the Young in Oryol at the 5th International Conference Science of the Future.

The results of the study are published in the article Rybalko A., Fradkov A. Identification of Two-Neuron FitzHugh-Nagumo Model Based on the Speed-Gradient and Filtering. Chaos. Vol. 33. Is. 8.https://doi.org/10.1063/5.0159132